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Bounds on the real stability radius for affinely perturbed differential‐algebraic systems
Author(s) -
Bracke M.,
PrätzelWolters D.
Publication year - 2002
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/1617-7061(200203)1:1<47::aid-pamm47>3.0.co;2-a
Subject(s) - mathematics , parameterized complexity , algebraic number , upper and lower bounds , differential algebraic equation , perturbation (astronomy) , ode , numerical stability , ordinary differential equation , stability (learning theory) , mathematical analysis , differential equation , numerical analysis , physics , combinatorics , quantum mechanics , computer science , machine learning
Analysis of robust stability for a family ( E ( Δ ), A ( Δ )) of linear differential‐algebraic equations (DAEs) depending on perturbations Δ ∈ Δ of some parameters is more difficult than for the classical ODE‐case where E(·) can be identified with I n ∈ ℝ n × n . We start with an electric circuit example for motivation. Then, after defining the class of parameterized DAEs we are dealing with we consider two kinds of stability radii: One concerns preservation of the structure for the perturbed system (including the algebraic index and dimension of the subspace belonging to the finite spectrum of ( E (·), A (·))). The second cares for stability of the finite spectrum as known from the classical case. Both can be treated independently and their combination yields the stability radius of the family. From this, it is possible to derive characterizations of both stability radii which are based on the structured singular value (SSV). However, the upper bounds may be very conservative in the real perturbation case – thus we introduce a variational principle which also characterizes the stability radius and allows for the computation of better upper bounds in the real perturbation case. In combination with the SSV‐based method this yields quite small intervals for the stability radius to lie in. Finally, some numerical results for the electric circuit example are presented.